A Variable Stepsize Implementation for Stochastic Differential Equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge–Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge–Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.